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Problem DDice and Ladders

The ladder game is a fun children’s game, the rules are as
follows: You start at cell number 1 and each round you roll a
dice and move the number specified by the dice. If you end on a
cell with a ladder starting from this cell then you have to
follow the ladder in its direction a single time, that is if
the ladder ends at a cell where a new ladder starts you will
not follow the new ladder. The game ends when you move to or
past the last cell.

Your task is to find the minimum number of dice-rolls
required to finish the game with a probability of at least
$p$.

Input

The first line contains three integers $r$ ($3
\le r \le 8$), $c$
($3 \le c \le 8$) and
$k$ ($0 \le k \le 50$) on one line, the
number of rows, columns and ladders respectively. The second
line contains a single floating point number $p$ ($0
< p < 1$) as described above (with at most
$6$ digits after the
decimal point).

Then follows $k$ lines,
each describing a ladder. The $i$’th of these lines contains two
integers $s_ i$
($2 \leq s_ i < r\cdot
c$) and $e_ i$
($1 \le e_ i \leq r\cdot
c$), the starting cell and ending cell of the ladder
$i$, respectively. Two
ladders will never start at the same cell, but multiple ladders
may end at the same cell. The cells are numbered like in the
illustration, meaning cell $1$ is in the bottom left corner and
there are $c$ more cells
in the same row. Cell $c+1$ starts to the left in the second
row, and so on.

It is guaranteed that it is possible to finish the game with
a probability of $p$ in
less than $10^8$
dice-rolls. The input is also constructed in such a way that
the expected number of dice-rolls such that you finish the game
with a probability of $p$
is the same as the expected number of dice-rolls such that you
finish the game with a probability of $p \pm 10^{-9}$

Output

A single integer, the minimum number of dice-rolls required
such that you finish the game with a probability at least
$p$.